Answer
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Work Step by Step
If a 2 by 2 matrix had more than 2 eigenvalues, it would have more than 2 linearly independent eigenvectors. However, because there are 2 rows and columns in a 2 by 2 matrix, any set of 3 vectors in $R^2$ is linearly dependent, meaning they cannot be eigenvectors.
Similarly, any set of m>n vectors in $R^n$ are linearly dependent, so there cannot be more than n eigenvalues with corresponding eigenvectors
